Experiment 9: Specific Heat Capacity and Calorimetry

Experiment 9: Specific Heat Capacity and Calorimetry

$Learning Objectives$

Explain temperature, heat, heat capacity, specific heat capacity, thermal equilibrium, and conservation of energy in a calorimetry experiment.
Determine the specific heat capacity of a heated solid sample using the method of mixtures.
Apply a calorimeter heat-capacity correction when a calorimeter constant is known or measured.
Distinguish the measured bath temperature from the sample temperature at the instant of transfer.
Quantify percent error relative to an appropriate reference value while recognizing that alloy composition affects reference data.
Identify systematic errors caused by heat loss during transfer, heat exchange with the calorimeter, evaporation, incomplete mixing, and thermometer response.
Design repeat trials and data checks that improve the reliability of the experimental result.

Calorimetry determines heat transfer from measured mass and temperature changes. In this experiment, a solid sample is heated in a water bath, transferred into cooler water in an insulated calorimeter, and allowed to reach thermal equilibrium. Conservation of energy then relates the heat lost by the hot sample to the heat gained by the water and calorimeter. The quality of the result depends strongly on rapid transfer, accurate masses, correct initial temperatures, and a well-defined equilibrium temperature.

$Target Learning Outcome$

Apply conservation of energy to a calorimeter system and determine the specific heat capacity of a material from measured masses and temperature changes while evaluating the principal sources of experimental error.

I. Discussion of Theory

$Temperature$

Temperature is a measure related to the average microscopic kinetic energy of particles and determines the direction of spontaneous heat transfer. Heat flows from a higher-temperature body toward a lower-temperature body until thermal equilibrium is reached.

$Heat$

Heat $Q$ is energy transferred because of a temperature difference. Heat is not a material stored inside an object; it is energy in transit across a system boundary.

$Specific Heat Capacity$

Specific heat capacity $c$ is the amount of energy required per unit mass to raise the temperature of a substance by one kelvin. A temperature interval of one kelvin has the same numerical size as an interval of one degree Celsius.

Sensible heat transfer

Use this expression when no phase change occurs and the specific heat is approximately constant over the temperature interval.

Q=mc\Delta T

Variables

Symbol	Description	Unit
$Q$	heat transferred to the body	J
$m$	mass	kg
$c$	specific heat capacity	J/(kg·K)
$\Delta T$	final temperature minus initial temperature	K or °C

$Sign convention for calorimetry$

If $Q=mc(T_f-T_i)$ is used, a body that warms has $Q>0$ and a body that cools has $Q<0$ . For an ideally insulated calorimeter system, the algebraic sum of heat transfers is zero:

\sum Q=0.

This sign convention prevents the common mistake of adding two positive heat quantities without accounting for which body loses energy.

$Thermal Equilibrium$

Thermal equilibrium is the state in which bodies in thermal contact have reached the same temperature and there is no net heat transfer between them.

$Heat Capacity$

Heat capacity $C$ is the energy required to raise the temperature of an entire object by one kelvin. It differs from specific heat capacity because it already includes the object's mass and composition.

Calorimeter heat

A calorimeter with heat capacity Ccal absorbs or releases energy as its temperature changes.

Q_{cal}=C_{cal}(T_f-T_{w,i})

Variables

Symbol	Description	Unit
$C_{cal}$	calorimeter heat capacity or calorimeter constant	J/K
$T_f$	equilibrium temperature	°C or K
$T_{w,i}$	initial water/calorimeter temperature	°C or K

$Energy balance for a hot solid placed in cooler water$

Let the hot sample have mass $m_s$ , unknown specific heat $c_s$ , and initial transfer temperature $T_{s,i}$ . Let the water have mass $m_w$ , known specific heat $c_w$ , and initial temperature $T_{w,i}$ . If the water and calorimeter begin at the same temperature and external heat exchange is negligible,

m_sc_s(T_f-T_{s,i})+m_wc_w(T_f-T_{w,i})+C_{cal}(T_f-T_{w,i})=0.

Rearranging for the sample specific heat gives

c_s=\frac{(m_wc_w+C_{cal})(T_f-T_{w,i})}{m_s(T_{s,i}-T_f)}.

Uncorrected specific heat

Use only when the calorimeter heat capacity is negligible or when the activity explicitly instructs you to neglect it.

c_s\approx\frac{m_wc_w(T_f-T_{w,i})}{m_s(T_{s,i}-T_f)}

$Use the sample transfer temperature, not automatically 100°C$

A metal heated in a boiling-water bath approaches the bath temperature but is not guaranteed to be exactly $100\,°C$ . Boiling temperature changes with atmospheric pressure, and the sample cools during transfer. Measure the bath temperature near the sample immediately before transfer and move the sample rapidly into the calorimeter. If a temperature probe can directly measure the sample, use that reading instead.

Calorimeter Calibration

$Why a calorimeter correction is needed$

An insulated cup, lid, thermometer, stirrer, and inner vessel can absorb a non-negligible amount of heat. If their heat capacity is ignored, the energy gained by the calorimeter is incorrectly attributed to the water, biasing the calculated sample specific heat.

Calorimeter constant from hot-water/cold-water mixing

One common calibration mixes known masses of warm and cool water in the calorimeter. This form assumes the calorimeter initially has the same temperature as the cool water.

C_{cal}=\frac{m_hc_w(T_{h,i}-T_f)-m_cc_w(T_f-T_{c,i})}{T_f-T_{c,i}}

Variables

Symbol	Description	Unit
$m_h$	mass of hot water	kg
$m_c$	mass of cool water initially in calorimeter	kg
$T_{h,i}$	initial hot-water temperature	°C
$T_{c,i}$	initial cool-water/calorimeter temperature	°C

$Reference value for water$

For introductory calorimetry near room temperature, $c_w\approx4184\,\text{J/(kg·K)}$ is commonly used. If the laboratory provides a different value or temperature-dependent property table, use the assigned value consistently.

Representative Specific Heat Values

Material	Representative specific heat near room temperature, J/(kg·K)	Caution
Aluminum	about 900	Depends on alloy and temperature.
Copper	about 385	High-purity values differ slightly with temperature.
Brass	about 370–390	Brass composition varies substantially.
Carbon steel	about 450–500	Depends on composition and temperature.

$Do not identify an unknown material from a single rough value alone$

Specific heat values overlap among alloys and vary with temperature. Use calorimetry to estimate a material property, not as a unique chemical-identification test unless supported by additional measurements.

II. Equipment and Materials

Equipment or material	Laboratory use
Insulated calorimeter with lid	Reduces heat exchange with the surroundings.
Metal test sample	Material whose specific heat is to be determined.
Digital balance	Measures sample and water masses.
Thermometer or temperature probe	Measures bath, initial-water, and equilibrium temperatures.
Beaker or heating vessel	Holds the hot-water bath.
Hot plate or laboratory heater	Heats the water bath.
Tongs, wire basket, or heat-safe thread	Transfers the heated sample safely and quickly.
Stirrer	Promotes uniform calorimeter temperature.
Graduated cylinder	Optional rough water-volume measurement; mass from a balance is preferred for calculations.
Paper towel	Dries liquid clinging to the exterior of the hot sample before transfer.

$Thermal and electrical safety$

Hot water, heated metal, steam, glassware, and heating surfaces can cause burns. Use tongs or heat-resistant handling tools. Keep electrical cords and probes dry, stabilize the hot-water vessel, and do not seal a heated vessel. Follow the laboratory's prescribed shutdown and spill procedures.

III. Pre-Laboratory Checks

$Before heating the sample$

Verify that the balance is level, clean, and zeroed.
Inspect the calorimeter for residual water and dry it before weighing.
Check thermometer/probe operation and confirm the displayed unit.
Identify the sample and record its mass before heating.
Ensure the water bath will fully cover the sample without the sample contacting a heating element.
Prepare the transfer path so the hot sample can move directly from bath to calorimeter without delay.
Place the calorimeter lid, thermometer, and stirrer within reach before removing the sample from the bath.

IV. Experimental Procedure

$Part A — Prepare the calorimeter and sample$

STEP-BY-STEP

Measure and record the dry calorimeter mass if your laboratory requires a mass-by-difference method.
Add enough room-temperature water to fully cover the metal sample after transfer.
Determine the water mass directly on a balance or by subtracting the dry calorimeter mass from the calorimeter-plus-water mass.
Insert the thermometer or probe, stir gently, and record the stable initial water temperature $T_{w,i}$ .
Measure and record the metal sample mass $m_s$ .
Place the sample in the hot-water bath. Keep it fully submerged for enough time to approach the bath temperature.
Stir the bath carefully so the temperature is reasonably uniform. Immediately before transfer, record the bath temperature near the sample as the best available estimate of $T_{s,i}$ .

$Part B — Transfer the sample and measure equilibrium temperature$

STEP-BY-STEP

Remove the sample with tongs, a basket, or heat-safe thread. Work quickly but safely.
Briefly remove excess bath water from the sample surface without allowing the sample to cool significantly. Water carried into the calorimeter changes the effective water mass and adds hot-water energy that is not included in the simple model.
Transfer the sample directly into the calorimeter and immediately replace the lid.
Stir gently and continuously without splashing. Keep the thermometer/probe from contacting the hot metal directly unless the instrument is intended for that measurement.
Record temperature at short, regular intervals. Identify the highest stable mixed temperature or use the laboratory's specified extrapolation method if a cooling curve is collected.
Record the equilibrium estimate as $T_f$ .
Verify that the physically required ordering is $T_{w,i}<T_f<T_{s,i}$ . If not, inspect the measurements before calculating $c_s$ .

$Part C — Calculate the sample specific heat$

STEP-BY-STEP

Use the corrected equation with the measured calorimeter constant if $C_{cal}$ is available.
If no calorimeter constant is supplied and the activity explicitly assumes a negligible calorimeter heat capacity, calculate the uncorrected value and state that assumption.
Check units before substitution: masses must be in kilograms when $c$ is in J/(kg·K).
Calculate the sample temperature drop $T_{s,i}-T_f$ and the water temperature rise $T_f-T_{w,i}$ separately before substituting.
Compute $c_s$ and retain reasonable significant figures based on the least precise measurements.
Compare with a defensible reference range for the known sample material and compute percent error.
Repeat the experiment for at least three trials if time permits. Use fresh initial conditions rather than immediately reusing hot calorimeter water.

$Part D — Optional calorimeter calibration by water mixing$

STEP-BY-STEP

Place a measured mass $m_c$ of cool water in the calorimeter and record $T_{c,i}$ after the calorimeter reaches the same temperature.
Prepare a measured mass $m_h$ of warmer water and record $T_{h,i}$ immediately before mixing.
Add the warm water to the calorimeter, replace the lid, stir, and record the equilibrium temperature $T_f$ .
Apply the calibration equation to determine $C_{cal}$ .
Repeat the calibration and compare values. A negative calculated heat capacity is physically impossible and indicates inconsistent measurements or an invalid thermal-history assumption.

V. Data and Results

Table 9.1 — Sample and Calorimeter Data

Quantity	Symbol	Unit
Sample material	—	—
Sample mass	$m_s$	kg
Water mass	$m_w$	kg
Initial sample/bath temperature	$T_{s,i}$	°C
Initial water temperature	$T_{w,i}$	°C
Equilibrium temperature	$T_f$	°C
Calorimeter heat capacity, if known	$C_{cal}$	J/K

Table 9.2 — Temperature-Time Observations

Time after transfer, s	Temperature, °C	Observation
0		Transfer completed
10
20
30
40
50
60

Table 9.3 — Specific Heat Results

Trial	$m_s$	$m_w$	$T_{s,i}$	$T_{w,i}$	$T_f$	$C_{cal}$	Computed $c_s$	Reference $c$	Percent error
1
2
3

Table 9.4 — Optional Calorimeter Calibration

Trial	$m_c$	$T_{c,i}$	$m_h$	$T_{h,i}$	$T_f$	Calculated $C_{cal}$ , J/K
1
2

VI. Computation and Data-Quality Requirements

$Required checks before accepting a result$

Confirm $T_{w,i}<T_f<T_{s,i}$ for a hot sample placed in cooler water.
Confirm every mass is converted to kilograms when using J/(kg·K).
Show the sample heat loss and water-plus-calorimeter heat gain with consistent signs.
State whether the calorimeter correction was included or intentionally neglected.
Verify the computed $c_s$ is positive and of a physically plausible order of magnitude.
Compare repeat trials before averaging; investigate a trial that differs substantially from the others.
Report percent error only against a reference value appropriate to the actual material or alloy.

VII. Experimental Uncertainty

$Dominant error mechanisms$

Cooling during transfer: The sample enters the calorimeter cooler than the measured bath temperature. Using the higher bath temperature in the denominator generally biases the calculated $c_s$ downward.
Hot water carried with the sample: Droplets from the heating bath add both mass and thermal energy that are absent from the simple energy balance.
Calorimeter heat capacity: Neglecting the cup, lid, probe, and stirrer assigns too little heat gain to the cold side of the system.
Heat exchange with the room: The system starts losing heat to the environment immediately after mixing. Delayed temperature readings usually underestimate the adiabatic equilibrium temperature.
Incomplete mixing: A thermometer located in a warm or cool region can record a local temperature rather than the bulk equilibrium value.
Thermometer lag: A sensor with slow response may miss the true peak mixed temperature.
Mass errors: Water splashed out or left on the sample changes the actual mass participating in the energy balance.

Direction-of-Bias Reasoning

$Why error direction matters$

A useful laboratory discussion does more than list possible errors. It predicts their effect. For example, if the sample cools before entering the calorimeter but the hotter bath temperature is still used as $T_{s,i}$ , the assumed sample temperature drop is too large. The calculated $c_s$ must then be smaller to account for the observed water warming. This produces a systematic low bias.

VIII. Post-Laboratory Questions

$Analysis questions$

Derive the corrected equation for $c_s$ from $\sum Q=0$ .
Explain why a calorimeter constant has units of J/K rather than J/(kg·K).
Explain why the sample should be transferred quickly but safely.
Predict the direction of error if the final temperature is recorded several minutes after mixing rather than near the initial equilibrium peak.
Predict the direction of error if hot bath water remains on the sample during transfer.
Explain why the water and calorimeter can be combined into the term $(m_wc_w+C_{cal})(T_f-T_{w,i})$ when both start at the same temperature.
Explain why Celsius temperature differences can be used in $Q=mc\Delta T$ even though the SI thermodynamic temperature unit is kelvin.
Compare the precision expected from one trial with the reliability obtained from several independent trials.

IX. Conclusion Guide

$Your conclusion should report$

the sample material and mean experimental specific heat;
whether a calorimeter correction was included;
the reference value or range used for comparison and the resulting percent error;
the consistency of repeated trials;
the most important systematic error and its likely direction of bias;
one concrete procedural improvement that would reduce that error.

Key Takeaways

Calorimetry is an energy-balance experiment: heat lost by hotter bodies equals heat gained by cooler bodies only when the chosen system is effectively insulated.
The sample transfer temperature, water mass, equilibrium temperature, and calorimeter heat capacity directly control the calculated specific heat.
Calorimeter correction is conceptually a heat-capacity term, not an additional mass of water unless a water-equivalent representation is explicitly defined.
Rapid transfer, complete mixing, and prompt temperature measurement are essential to reduce systematic heat-loss error.
A credible laboratory result includes uncertainty reasoning and repeatability, not only a numerical value close to a handbook reference.

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